Galerkin Solutions Of Bend Of A Rectangular Thin Plate With Four Free Edges On Gibson Foudation

Plates on elastic foundation have been widely used in engineering,and many engineering structures can be simplified into the elastic plates.Therefore,it is necessary to accurately and reasonably calculate the plates.Although there are many elastic foundation models and they all have their own characteristics,Two-parameter foundation model has irreplaceable advantages.In this paper,by comparing the different elastic foundations,the Vlasov Two-parameter foundation model which is simple and can fully describe the elastic foundation is chosen as the foundation model of this paper.It is worth noting that the assumption of homogeneous soil is an important prerequisite in the classical mechanics theory.However,most of the foundation soil in engineering practice is non-uniform.In other words,some physical properties of foundation soil will change along with the increase of foundation depth,which leads to the limitation of the classical mechanics theory based on the homogeneous soil hypothesis.In engineering practice,the foundation is often considered as the Gibson foundation,that is to say,the elastic modulus of the soil changes linearly along the depth of the foundation.In this paper,based on the Vlasov Two-parameter foundation model,the elastic modulus of the foundation soil is considered as a quantity that varies linearly along the depth of the foundation,so that the bending problem of a rectangular thin plate with four defined free edges on the Gibson foudation is solved.The result of this improvement shows that the non-uniformity of the soil has a great influence on internal forces and displacement of a plate which is on elastic foundation and should be considered.Based on the principle of minimum potential energy,this paper uses the variationalmethod to derive the control differential equations and boundary conditions of rectangular free plates on the Gibson foundation model under load.The key to solving this differential equation is to find a suitable deflection function,and then use a corresponding method to solve the problem.In this paper,by comparing different trial functions,a deflection function that satisfies the boundary conditions of the displacement and satisfies the boundary conditions of the force is found.Using Galerkin method,the total residual of the control equation is integrated into zero in the whole board domain,so that each undetermined coefficients of the selected trial function are obtained,and an exact function which can describe displacement of each point of the rectangular plate on the Gibson foundation model under load is obtained.Through the theoretical derivation and programming of related problems,the Galerkin solutions of a four-sided free rectangular plate on the Gibson foundation is obtained.Through comparing with previous examples,the convergence and correctness of the results are analyzed.At the same time,the influence of various parameters on the bending of thin plates on the foundation is also studied.The research method takes into account the effects of soil non-uniformity on the internal forces and displacements of the plate,which are ignored by previous researchers,and can be applied to the solution of various related problems.The research method has a progressive significance.